Let $\mathcal{C}$ be a site. Recall that a semi-representable object of $\mathcal{C}$ is simply a family $\{ U_ i\} _{i \in I}$ of objects of $\mathcal{C}$. A morphism $\{ U_ i\} _{i \in I} \to \{ V_ j\} _{j \in J}$ of semi-representable objects is given by a map $\alpha : I \to J$ and for every $i \in I$ a morphism $f_ i : U_ i \to V_{\alpha (i)}$ of $\mathcal{C}$. The category of semi-representable objects of $\mathcal{C}$ is denoted $\text{SR}(\mathcal{C})$. See Hypercoverings, Definition 25.2.1 and the enclosing section for more information.
For a semi-representable object $K = \{ U_ i\} _{i \in I}$ of $\mathcal{C}$ we let
be the disjoint union of the localizations of $\mathcal{C}$ at $U_ i$. There is a natural structure of a site on this category, with coverings inherited from the localizations $\mathcal{C}/U_ i$. The site $\mathcal{C}/K$ is called the localization of $\mathcal{C}$ at $K$. Observe that a sheaf on $\mathcal{C}/K$ is the same thing as a family of sheaves $\mathcal{F}_ i$ on $\mathcal{C}/U_ i$, i.e.,
This is occasionally useful to understand what is going on.
Let $\mathcal{C}$ be a site. Let $K = \{ U_ i\} _{i \in I}$ be an object of $\text{SR}(\mathcal{C})$. There is a continuous and cocontinuous localization functor $j : \mathcal{C}/K \to \mathcal{C}$ which is the product of the localization functors $j_ i : \mathcal{C}/V_ i \to \mathcal{C}$. We obtain functors $j_!$, $j^{-1}$, $j_*$ exactly as in Sites, Section 7.25. In terms of the product decomposition $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K) = \prod \nolimits _{i \in I} \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_ i)$ we have
as the reader easily verifies.
Let $f : K \to L$ be a morphism of $\text{SR}(\mathcal{C})$. Then we obtain a continuous and cocontinuous functor
by applying the construction of Sites, Lemma 7.25.8 to the components. More precisely, suppose $f = (\alpha , f_ i)$ where $K = \{ U_ i\} _{i \in I}$, $L = \{ V_ j\} _{j \in J}$, $\alpha : I \to J$, and $f_ i : U_ i \to V_{\alpha (i)}$. Then the functor $v$ maps the component $\mathcal{C}/U_ i$ into the component $\mathcal{C}/V_{\alpha (i)}$ via the construction of the aforementioned lemma. In particular we obtain a morphism
of topoi. In terms of the product decompositions $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K) = \prod \nolimits _{i \in I} \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_ i)$ and $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/L) = \prod \nolimits _{j \in J} \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/V_ j)$ the reader verifies that
where $f_ i : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_ i) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/V_{\alpha (i)})$ is the morphism associated to the localization functor $\mathcal{C}/U_ i \to \mathcal{C}/V_{\alpha (i)}$ corresponding to $f_ i : U_ i \to V_{\alpha (i)}$.
Lemma 85.15.1. Let $\mathcal{C}$ be a site.
For $K$ in $\text{SR}(\mathcal{C})$ the functor $j : \mathcal{C}/K \to \mathcal{C}$ is continuous, cocontinuous, and has property P of Sites, Remark 7.20.5.
For $f : K \to L$ in $\text{SR}(\mathcal{C})$ the functor $v : \mathcal{C}/K \to \mathcal{C}/L$ (see above) is continuous, cocontinuous, and has property P of Sites, Remark 7.20.5.
Proof. Proof of (2). In the notation of the discussion preceding the lemma, the localization functors $\mathcal{C}/U_ i \to \mathcal{C}/V_{\alpha (i)}$ are continuous and cocontinuous by Sites, Section 7.25 and satisfy $P$ by Sites, Remark 7.25.11. It is formal to deduce $v$ is continuous and cocontinuous and has $P$. We omit the details. We also omit the proof of (1). $\square$
Lemma 85.15.2. Let $\mathcal{C}$ be a site and $K$ in $\text{SR}(\mathcal{C})$. For $\mathcal{F}$ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ we have where $F$ is as in Hypercoverings, Definition 25.2.2.
\[ j_*j^{-1}\mathcal{F} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits (F(K)^\# , \mathcal{F}) \]
Proof. Say $K = \{ U_ i\} _{i \in I}$. Using the description of the functors $j^{-1}$ and $j_*$ given above we see that
The second equality by Sites, Lemma 7.26.3. Since $F(K) = \coprod h_{U_ i}$ in $\textit{PSh}(\mathcal{C}$, we have $F(K)^\# = \coprod h_{U_ i}^\# $ in $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ and since $\mathop{\mathcal{H}\! \mathit{om}}\nolimits (-, \mathcal{F})$ turns coproducts into products (immediate from the construction in Sites, Section 7.26), we conclude. $\square$
Lemma 85.15.3. Let $\mathcal{C}$ be a site.
For $K$ in $\text{SR}(\mathcal{C})$ the functor $j_!$ gives an equivalence $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/F(K)^\# $ where $F$ is as in Hypercoverings, Definition 25.2.2.
The functor $j^{-1} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K)$ corresponds via the identification of (1) with $\mathcal{F} \mapsto (\mathcal{F} \times F(K)^\# \to F(K)^\# )$.
For $f : K \to L$ in $\text{SR}(\mathcal{C})$ the functor $f^{-1}$ corresponds via the identifications of (1) to the functor $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/F(L)^\# \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})/F(K)^\# $, $(\mathcal{G} \to F(L)^\# ) \mapsto (\mathcal{G} \times _{F(L)^\# } F(K)^\# \to F(K)^\# )$.
Proof. Observe that if $K = \{ U_ i\} _{i \in I}$ then the category $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K)$ decomposes as the product of the categories $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_ i)$. Observe that $F(K)^\# = \coprod _{i \in I} h_{U_ i}^\# $ (coproduct in sheaves). Hence $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/F(K)^\# $ is the product of the categories $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})/h_{U_ i}^\# $. Thus (1) and (2) follow from the corresponding statements for each $i$, see Sites, Lemmas 7.25.4 and 7.25.7. Similarly, if $L = \{ V_ j\} _{j \in J}$ and $f$ is given by $\alpha : I \to J$ and $f_ i : U_ i \to V_{\alpha (i)}$, then we can apply Sites, Lemma 7.25.9 to each of the re-localization morphisms $\mathcal{C}/U_ i \to \mathcal{C}/V_{\alpha (i)}$ to get (3). $\square$
Lemma 85.15.4. Let $\mathcal{C}$ be a site. For $K$ in $\text{SR}(\mathcal{C})$ the functor $j^{-1}$ sends injective abelian sheaves to injective abelian sheaves. Similarly, the functor $j^{-1}$ sends K-injective complexes of abelian sheaves to K-injective complexes of abelian sheaves.
Proof. The first statement is the natural generalization of Cohomology on Sites, Lemma 21.7.1 to semi-representable objects. In fact, it follows from this lemma by the product decomposition of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K)$ and the description of the functor $j^{-1}$ given above. The second statement is the natural generalization of Cohomology on Sites, Lemma 21.20.1 and follows from it by the product decomposition of the topos.
Alternative: since $j$ induces a localization of topoi by Lemma 85.15.3 part (1) it also follows immediately from Cohomology on Sites, Lemmas 21.7.1 and 21.20.1 by enlarging the site; compare with the proof of Cohomology on Sites, Lemma 21.13.3 in the case of injective sheaves. $\square$
Remark 85.15.5 (Variant for over an object). Let $\mathcal{C}$ be a site. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. The category $\text{SR}(\mathcal{C}, X)$ of semi-representable objects over $X$ is defined by the formula $\text{SR}(\mathcal{C}, X) = \text{SR}(\mathcal{C}/X)$. See Hypercoverings, Definition 25.2.1. Thus we may apply the above discussion to the site $\mathcal{C}/X$. Briefly, the constructions above give
a site $\mathcal{C}/K$ for $K$ in $\text{SR}(\mathcal{C}, X)$,
a decomposition $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K) = \prod \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/U_ i)$ if $K = \{ U_ i/X\} $,
a localization functor $j : \mathcal{C}/K \to \mathcal{C}/X$,
a morphism $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}/L)$ for $f : K \to L$ in $\text{SR}(\mathcal{C}, X)$.
All results of this section hold in this situation by replacing $\mathcal{C}$ everywhere by $\mathcal{C}/X$.
Remark 85.15.6 (Ringed variant). Let $\mathcal{C}$ be a site. Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings on $\mathcal{C}$. In this case, for any semi-representable object $K$ of $\mathcal{C}$ the site $\mathcal{C}/K$ is a ringed site with sheaf of rings $\mathcal{O}_ K = j^{-1}\mathcal{O}_\mathcal {C}$. The constructions above give
a ringed site $(\mathcal{C}/K, \mathcal{O}_ K)$ for $K$ in $\text{SR}(\mathcal{C})$,
a decomposition $\textit{Mod}(\mathcal{O}_ K) = \prod \textit{Mod}(\mathcal{O}_{U_ i})$ if $K = \{ U_ i\} $,
a localization morphism $j : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K), \mathcal{O}_ K) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C})$ of ringed topoi,
a morphism $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K), \mathcal{O}_ K) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/L), \mathcal{O}_ L)$ of ringed topoi for $f : K \to L$ in $\text{SR}(\mathcal{C})$.
Many of the results above hold in this setting. For example, the functor $j^*$ has an exact left adjoint
which in terms of the product decomposition given in (2) sends $(\mathcal{F}_ i)_{i \in I}$ to $\bigoplus j_{i, !}\mathcal{F}_ i$. Similarly, given $f : K \to L$ as above, the functor $f^*$ has an exact left adjoint $f_! : \textit{Mod}(\mathcal{O}_ K) \to \textit{Mod}(\mathcal{O}_ L)$. Thus the functors $j^*$ and $f^*$ are exact, i.e., $j$ and $f$ are flat morphisms of ringed topoi (also follows from the equalities $\mathcal{O}_ K = j^{-1}\mathcal{O}_\mathcal {C}$ and $\mathcal{O}_ K = f^{-1}\mathcal{O}_ L$).
Remark 85.15.7 (Ringed variant over an object). Let $\mathcal{C}$ be a site. Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings on $\mathcal{C}$. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and denote $\mathcal{O}_ X = \mathcal{O}_\mathcal {C}|_{\mathcal{C}/U}$. Then we can combine the constructions given in Remarks 85.15.5 and 85.15.6 to get
a ringed site $(\mathcal{C}/K, \mathcal{O}_ K)$ for $K$ in $\text{SR}(\mathcal{C}, X)$,
a decomposition $\textit{Mod}(\mathcal{O}_ K) = \prod \textit{Mod}(\mathcal{O}_{U_ i})$ if $K = \{ U_ i\} $,
a localization morphism $j : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K), \mathcal{O}_ K) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X), \mathcal{O}_ X)$ of ringed topoi,
a morphism $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K), \mathcal{O}_ K) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/L), \mathcal{O}_ L)$ of ringed topoi for $f : K \to L$ in $\text{SR}(\mathcal{C}, X)$.
Of course all of the results mentioned in Remark 85.15.6 hold in this setting as well.
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